Lecture 5

Question 1

how does an evolution strategy work?

Answer 1

• initialize the population
• evaluate the population
  WHILE termination criterion not met DO:
• recombination
• mutation
• selection ((mu + lambda) or (mu, lambda))
• evaluate the new generation
• set the population to the new generation

Question 2

what kinds of mutation operators exist for ES?

Answer 2

• one sigma (self-adaptive with one step size)
• individual sigma (self-adaptive with individual step sizes)
• correlated mutations

Question 3

how does one sigma work?

Answer 3

individual before mutation: a = ((x1, x2,…), sigma)
mutate step size: sigma’ = sigma exp(tau_0*N(0,1))
mutate search variables: x’_i = x_i + sigma’ * N(0,1)
individual after mutation: a’=((x’1, x’2, …), sigma’)

Question 4

what is tau_0?

Answer 4

it’s the learning rate
- affects speed of the sigma adaptation
- tau_0 bigger: faster but less precise
- tau_0 smaller: slower but more precise
- according to Schwefel: tau_0 = 1/sqrt(n)

Question 5

what are the pros and cons of one sigma?

Answer 5

pros:
- simple adaptation mechanism
- self-adaptation usually fast and precise

cons:
- bad adaptation in case of complicated contour lines
- bad adaptation in case of very differently scaled object variables

Question 6

how does individual mutation work?

Answer 6

before: a = ((x1, x2,…), (sigma1, sigma2))
1. sample a gobal perturbation: g ~ N(0,1)
2. mutate individual step sizes:
sigma’_i = sigma_i exp(tau’g + tauN(0,1))
3. mutate search variables:
x’_i = x_i + sigma’_i * N(0,1)
after: ((x’1, x’2,…), (sigma’1, sigma’2))

Question 7

what are tau and tau’?

Answer 7

tau’ is the global learning rate
tau is the local learning rate
Schwefel: tau’ = 1/sqrt(2n) and
tau = 1/sqrt(2*sqrt(n))

Question 8

what are the pros and cons of individual mutation?

Answer 8

pros:
- individual scaling of object variables
- increased global convergence reliability

cons:
- slower convergence due to increased learning effort
- no rotation of coordinate system possible

Question 9

What is correlated mutation?

Answer 9

• changes to the parameters are not independent, but guided by a covariance matrix that captures correlation between the different elements of the solution vector
• also the rotation angles of planes are adapted
• the covariance matrix is obtained by multiplying the solution vector by the rotation matrix
• the correlation matrix must be positive definitive

Question 10

what are the pros and cons of correlated mutation?

Answer 10

pros:
- individual scaling of object variables
- rotation of coordinate system possible
- increased global convergence reliability

cons:
- much slower convergence
- effort for mutations scales quadratically
- self-adaptation very inefficient

Question 11

what is recombination?

Answer 11

the step in an ES directly after selection that iteratively generates lambda offspring

Question 12

how does discrete recombination work?

Answer 12

the variable at position i will be copied at random (uniformly distributed) from position i of parent1 or parent2 to the offspring

Question 13

how does intermediate recombination work?

Answer 13

the variable at position i of the offspring is the artithmetic mean of position i from parent1 and parent2

Question 14

how does global discrete recombination work?

Answer 14

consider all parents and copy one of the parents at position i randomly to position i of the offpsring

Question 15

how does global intermediary recombination work?

Answer 15

take the mean of all parents at position i

Question 16

what is (mu + lambda) selection?

Answer 16

• mu parents produce lambda offspring by (recombination and) mutation
• the mu best out of mu + lambda will be selected (deterministic selection)
• this method guarantees monotonicity: deteriorations will never be accepted

Question 17

what is (mu, lambda) selection?

Answer 17

• mu parents produce lambda&raquo_space; mu offspring by (recombination and) mutation
• the mu best out of lambda offspring will be selected
• this method does not guarantee monotonicity: deteriorations are possible

Question 18

what are the possible occurrences of selection in ES?

Answer 18

• (1+1): one parent, one offspring, 1/5 rule
• (1, lambda): one parent, lambda offspring
• (mu, lambda): mu > 1 parents, lambda > mu offspring (can overcome local optima)
• elitist strategies

Question 19

what is the takeover time tau*?

Answer 19

the number of generations until repeated application of selection completely fills the population with copies of the initially best individual

for (mu, lambda) selection in ES:
tau* = ln lambda / ln (lambda/mu)

for proportional selection in GA:
tau* = lambda ln lambda

Question 20

what are the update rules for correlated mutation?

Answer 20

sigma update same as individual sigma
alpha’_j = alpha_j + N(0, beta) with j in [n(n-1)/2]
x’_i = x_i + sigma’_i * N(0, C’)
with C’ (the new covariance matrix) computed from alpha’_j and sigma’_i